scholarly journals On the Canonical Foliation of an Indefinite Locally Conformal Kähler Manifold with a Parallel Lee Form

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 333
Author(s):  
Elisabetta Barletta ◽  
Sorin Dragomir ◽  
Francesco Esposito
Keyword(s):  
Decomposition Theorem ◽  
Conformally Flat ◽  
Canonical Foliation ◽  
Locally Conformal Kähler ◽  
De Rham ◽  
Vaisman Manifold ◽  
Geodesically Complete ◽  
Locally Conformal ◽  
Lee Form ◽  
Hopf Manifold

We study the semi-Riemannian geometry of the foliation F of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation ω=0, provided that ∇ω=0 and c=∥ω∥≠0 (ω is the Lee form of M). If M is conformally flat then every leaf of F is shown to be a totally geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature c/4, carrying an indefinite c-Sasakian structure. As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem any geodesically complete, conformally flat, indefinite Vaisman manifold of index 2s, 0<s<n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold CHsn(λ), 0<λ<1, equipped with the indefinite Boothby metric gs,n.

scholarly journals Example of a six-dimensional LCK solvmanifold

2017 ◽  
Vol 4 (1) ◽  
pp. 37-42
Author(s):  
Hiroshi Sawai
Keyword(s):  
Lie Group ◽  
Locally Conformal Kähler ◽  
Locally Conformal ◽  
Lee Form ◽  
Solvable Lie Group

Abstract The purpose of this paper is to prove that there exists a lattice on a certain solvable Lie group and construct a six-dimensional locally conformal Kähler solvmanifold with non-parallel Lee form.

scholarly journals On locally conformal Kähler space forms

1985 ◽  
Vol 8 (1) ◽  
pp. 69-74 ◽  
Author(s):  
Koji Matsumoto
Keyword(s):  
Space Form ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Covariant Differentiation ◽  
Space Forms ◽  
Riemannian Curvature Tensor ◽  
Locally Conformal Kähler ◽  
Necessary And Sufficient ◽  
Locally Conformal ◽  
Lee Form

Anm-dimensional locally conformal Kähler manifold (l.c.K-manifold) is characterized as a Hermitian manifold admitting a global closedl-formαλ(called the Lee form) whose structure(Fμλ,gμλ)satisfies∇νFμλ=−βμgνλ+βλgνμ−αμFνλ+αλFνμ, where∇λdenotes the covariant differentiation with respect to the Hermitian metricgμλ,βλ=−Fλϵαϵ,Fμλ=Fμϵgϵλand the indicesν,μ,…,λrun over the range1,2,…,m.For l. c. K-manifolds, I. Vaisman [4] gave a typical example and T. Kashiwada ([1], [2],[3]) gave a lot of interesting properties about such manifolds.In this paper, we shall study certain properties of l. c. K-space forms. In§2, we shall mainly get the necessary and sufficient condition that an l. c. K-space form is an Einstein one and the Riemannian curvature tensor with respect togμλwill be expressed without the tensor fieldPμλ. In§3, we shall get the necessary and sufficient condition that the length of the Lee form is constant and the sufficient condition that a compact l. c. K-space form becomes a complex space form. In the last§4, we shall prove that there does not exist a non-trivial recurrent l. c. K-space form.

scholarly journals Infinitesimal Transformations of Locally Conformal Kähler Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 658
Author(s):  
Yevhen Cherevko ◽  
Volodymyr Berezovski ◽  
Irena Hinterleitner ◽  
Dana Smetanová
Keyword(s):  
Sufficient Conditions ◽  
Lie Derivative ◽  
Conformal Transformations ◽  
Projective Transformations ◽  
Integrability Conditions ◽  
Locally Conformal Kähler ◽  
Necessary And Sufficient ◽  
Locally Conformal ◽  
Lee Form ◽  
Infinitesimal Transformations

The article is devoted to infinitesimal transformations. We have obtained that LCK-manifolds do not admit nontrivial infinitesimal projective transformations. Then we study infinitesimal conformal transformations of LCK-manifolds. We have found the expression for the Lie derivative of a Lee form. We have also obtained the system of partial differential equations for the transformations, and explored its integrability conditions. Hence we have got the necessary and sufficient conditions in order that the an LCK-manifold admits a group of conformal motions. We have also calculated the number of parameters which the group depends on. We have proved that a group of conformal motions admitted by an LCK-manifold is isomorphic to a homothetic group admitted by corresponding Kählerian metric. We also established that an isometric group of an LCK-manifold is isomorphic to some subgroup of the homothetic group of the coresponding local Kählerian metric.

scholarly journals A NATURAL CONNECTION ON A BASIC CLASS OF RIEMANNIAN PRODUCT MANIFOLDS

2012 ◽  
Vol 09 (07) ◽  
pp. 1250057 ◽  
Author(s):  
DOBRINKA GRIBACHEVA
Keyword(s):  
Curvature Tensor ◽  
Sufficient Conditions ◽  
Flat Connection ◽  
Riemannian Product ◽  
Natural Connection ◽  
Hermitian Geometry ◽  
Locally Conformal Kähler ◽  
Basic Class ◽  
Necessary And Sufficient ◽  
Locally Conformal

A Riemannian manifold M with an integrable almost product structure P is called a Riemannian product manifold. Our investigations are on the manifolds (M, P, g) of the largest class of Riemannian product manifolds, which is closed with respect to the group of conformal transformations of the metric g. This class is an analogue of the class of locally conformal Kähler manifolds in almost Hermitian geometry. In the present paper we study a natural connection D on (M, P, g) (i.e. DP = Dg = 0). We find necessary and sufficient conditions, the curvature tensor of D to have properties similar to the Kähler tensor in Hermitian geometry. We pay attention to the case when D has a parallel torsion. We establish that the Weyl tensors for the connection D and the Levi-Civita connection coincide as well as the invariance of the curvature tensor of D with respect to the usual conformal transformation. We consider the case when D is a flat connection. We construct an example of the considered manifold by a Lie group where D is a flat connection with non-parallel torsion.

scholarly journals Examples of non-trivial rank in locally conformal Kähler geometry

2010 ◽  
Vol 270 (1-2) ◽  
pp. 179-187 ◽  
Author(s):  
Maurizio Parton ◽  
Victor Vuletescu
Keyword(s):  
Kahler Geometry ◽  
Locally Conformal Kähler ◽  
Locally Conformal
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